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Soft modes near the buckling transition of icosahedral shells
Outline:•Mass and spring model•Buckling transition•Eigenvalue spectrum of Hessian•Soft mode•Effective stiffness (vertex/edge/face)
with Jack Lidmar and David Nelson
Mass and Spring Model
a
ks
n1 n2
Es=(1/2)ksΣ(|r1-r2|-a)2
Eb=(1/2)kbΣ|n1-n2|2
kb
Dimensionless combination: ~ ks R2/kb
“Foppl-von Karman Number”2x 3x 5x
Buckling Transition(Lidmar, Mirny and Nelson 2003)
= 120 = 15000 = 230
Color indicates energypurple = zero
Color indicates Gaussian curvaturepurple = negative
Large limits:Vertex: Longitudinal deformation of triangular face K~aY/log(b) Edge: Ridge scaling (Witten et al.) in presence of force K~Y/1/2
Face: Axial force on residual curvature K~Y/2/3
Small limit:Response of sphere to isotropic pressure: NK~12Y
Medium :Soft mode concentrated on vertices